Approximation of the set of integrable trajectories of the control system with $L_2$ norm constraints on control functions

Abstract

In this paper an approximation of the set of multivariable and $L_2$ integrable trajectories of the control system described by Urysohn type integral equation is considered. It is assumed that the system is affine with respect to the control vector. The admissible control functions are chosen from the closed ball of the space $L_2$, centered at the origin with radius $\rho$. The set of admissible control functions is replaced, step by step, by the set of controls consisting of a finite number of piecewise-constant control functions. It is proved that under appropriate choosing of the discretization parameters, the set of trajectories generated by a finite number of piecewise-constant control functions is an internal approximation of the set of trajectories.